The question is

Let G be a group, such that

let with finite order show that

if

then

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- December 14th 2011, 07:03 AMAmerwhen a subgroup is in the center of other subgroup
The question is

Let G be a group, such that

let with finite order show that

if

then - December 14th 2011, 07:39 AMDevenoRe: when a subgroup is in the center of other subgroup
i am a little unsure as to how [G:N] = r is relevant to your question, other than saying N is of finite index. i also don't know what you mean by "one subgroup is in the center of the other" in terms of your subsequent question.

if H is in Z(N), of course nh = hn, but then it cannot be the case that H∩N = {e}, unless Z(N) is likewise trivial.

but if H,N are both normal in G, with H∩N = {e}, then for any h in H, and n in N, we have:

since , because N is normal, while

, since because H is normal.

since H∩N = {e}, we see that h and n commute. - December 14th 2011, 07:52 AMAmerRe: when a subgroup is in the center of other subgroup
thanks

the problem in Abstract algebra when you see the answer you say oooh it is easy how i did not figure that

the hardness is the idea of the solution

I think if I had thought in it in reverse nh = hn and see what will happen

anyway thanks very much